Properties

Label 4008.2.a
Level 4008
Weight 2
Character orbit a
Rep. character \(\chi_{4008}(1,\cdot)\)
Character field \(\Q\)
Dimension 82
Newforms 13
Sturm bound 1344
Trace bound 7

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Defining parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)
Character field: \(\Q\)
Newforms: \( 13 \)
Sturm bound: \(1344\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4008))\).

Total New Old
Modular forms 680 82 598
Cusp forms 665 82 583
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(167\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(11\)
\(+\)\(-\)\(+\)\(-\)\(14\)
\(+\)\(-\)\(-\)\(+\)\(7\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(9\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(37\)
Minus space\(-\)\(45\)

Trace form

\(82q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 82q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(82q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 82q^{9} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 16q^{19} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 86q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut +\mathstrut 24q^{35} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 82q^{49} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 16q^{55} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut -\mathstrut 32q^{65} \) \(\mathstrut -\mathstrut 28q^{67} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 12q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 24q^{79} \) \(\mathstrut +\mathstrut 82q^{81} \) \(\mathstrut +\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 24q^{85} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 16q^{93} \) \(\mathstrut -\mathstrut 32q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 167
4008.2.a.a \(1\) \(32.004\) \(\Q\) None \(0\) \(-1\) \(1\) \(-3\) \(-\) \(+\) \(+\) \(q-q^{3}+q^{5}-3q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
4008.2.a.b \(1\) \(32.004\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(+\) \(-\) \(-\) \(q+q^{3}-q^{5}-q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
4008.2.a.c \(1\) \(32.004\) \(\Q\) None \(0\) \(1\) \(0\) \(0\) \(+\) \(-\) \(-\) \(q+q^{3}+q^{9}+2q^{17}-8q^{19}-4q^{23}+\cdots\)
4008.2.a.d \(1\) \(32.004\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(+\) \(-\) \(+\) \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4008.2.a.e \(3\) \(32.004\) 3.3.148.1 None \(0\) \(-3\) \(6\) \(-4\) \(-\) \(+\) \(-\) \(q-q^{3}+(2-\beta _{2})q^{5}+(-2+2\beta _{1})q^{7}+\cdots\)
4008.2.a.f \(5\) \(32.004\) 5.5.284897.1 None \(0\) \(5\) \(1\) \(-4\) \(+\) \(-\) \(-\) \(q+q^{3}+(-\beta _{3}+\beta _{4})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
4008.2.a.g \(7\) \(32.004\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-7\) \(-3\) \(8\) \(-\) \(+\) \(+\) \(q-q^{3}-\beta _{1}q^{5}+(1+\beta _{2})q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4008.2.a.h \(8\) \(32.004\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-8\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q-q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}+q^{9}+(\beta _{2}+\beta _{4}+\cdots)q^{11}+\cdots\)
4008.2.a.i \(9\) \(32.004\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(9\) \(-6\) \(-11\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1+\beta _{1})q^{5}+(-2+\beta _{5}+\beta _{7}+\cdots)q^{7}+\cdots\)
4008.2.a.j \(10\) \(32.004\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-10\) \(-10\) \(1\) \(+\) \(+\) \(+\) \(q-q^{3}+(-1-\beta _{1})q^{5}+(\beta _{2}+\beta _{4}+\beta _{7}+\cdots)q^{7}+\cdots\)
4008.2.a.k \(11\) \(32.004\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(-11\) \(10\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{3}+(1-\beta _{1})q^{5}+\beta _{3}q^{7}+q^{9}-\beta _{10}q^{11}+\cdots\)
4008.2.a.l \(12\) \(32.004\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(12\) \(4\) \(11\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta _{1}q^{5}+(1-\beta _{9})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
4008.2.a.m \(13\) \(32.004\) \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(13\) \(2\) \(1\) \(+\) \(-\) \(+\) \(q+q^{3}+\beta _{1}q^{5}-\beta _{9}q^{7}+q^{9}+(1+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(668))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\)\(^{\oplus 2}\)